Susantha Goonatilake writes that the development of the Fibonacci sequence"is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD),Gopāla (c. 1135), and Hemachandra (c. 1150)".[7]

However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

The sequence is also discussed by Gopala (before 1135 AD) and by the Jainscholar Hemachandra (c. 1150).

​Golden Ratio

Fibonacci Numbers

The golden ratio, geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture.

Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra. For examples of sacred geometry in art and architecture refer: ​Labyrinth (an Eulerian path, as distinct from a maze) Mandala – Parthenon -Taijitu (Yin-Yang) - Tree of Life - Rose Window Celtic art such as the Book of Kells – Yantra - Dharmacakra

Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry.

The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence.

Since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.

Fibonacci Numbers

By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Some sources omit the initial 0, instead beginning the sequence with two 1s.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation with seed values.

The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci (a contraction of filius Bonacci, "son of Bonaccio"). Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence may have been previously described in Indian mathematics.

Fibonacci numbers are used in the analysis of financial markets, in strategies such as Fibonacci retracement, and are used in computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure.

The simple recursion of Fibonacci numbers has also inspired a family of recursive graphs called Fibonacci cubes for interconnecting parallel and distributed systems.​ They also appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone.

Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic. He soon realised the many advantages of the Hindu-Arabic system. In 1202, he completed the Liber Abaci (Book of Abacus or Book of Calculation) which popularized Hindu–Arabic numerals in Europe. [2]

Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. In 1240, the Republic of Pisa honored Fibonacci (referred to as Leonardo Bigollo)[8] by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens. [9]

The date of Fibonacci's death is not known, but it has been estimated to be between 1240 [10] and 1250, [11] most likely in Pisa.

​In the Liber Abaci (1202), Fibonacci introduced the so-called modus Indorum (method of the Indians), today known as Hindu-Arabic numerals. [12][13] The book advocated numeration with the digits 0–9 and place value. The book showed the practical use and value of the new Arabic numeral system by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, and other applications.

​The book was well-received throughout educated Europe and had a profound impact on European thought. No copies of the 1202 edition are known to exist. [14]

The 1228 edition, first section introduces the Arabic numeral system and compares the system with other systems, such as Roman numerals, and methods to convert the other numeral systems into Arabic numerals. Replacing the Roman numeral system, its ancient Egyptian multiplication method, and using an abacus for calculations, with an Arabic numeral system was an advance in making business calculations easier and faster, which led to the growth of banking and accounting in Europe. [15][16]

The second section explains the uses of Arabic numerals in business, for example converting different currencies, and calculating profit and interest, which were important to the growing banking industry. The book also discusses irrational numbers and prime numbers. [14][15][16]Fibonacci sequence.

Liber Abaci posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers.

Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been noted by Indian mathematicians as early as the sixth century. [17][18][19][20]​In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers.

Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1,1, 2, etc. He carried the calculation up to the thirteenth place (fourteenth in modern counting), that is 233, though another manuscript carries it to the next place:

Le Corbusier 's faith in the mathematical order of the universe was closely bound tothe golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another.

In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took Leonardo's suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system.[47]

Buildings by Le Corbusier:

The United Nations Headquarters, New York. 1947

Chapel of Notre Dame du Haut, Ronchamp, Belfort, France. 1954

The Open Hand (La Main Ouverte) is a Le Corbusier's architecture,

​a sign for him of.

"Peace and reconciliation,

it is open to give and open to receive."

The largest sculptures that Le Corbusier created is a 26 meter high version.